--- title: "Visualizing Distributions" output: html_document: toc: yes code_folding: show code_download: true --- ```{r setup, include = FALSE, message = FALSE} source(here::here("setup.R")) knitr::opts_chunk$set(collapse = TRUE, message = FALSE, fig.height = 5, fig.width = 6, fig.align = "center") set.seed(12345) library(dplyr) library(ggplot2) library(lattice) library(gridExtra) source(here::here("datasets.R")) ``` ## Introduction Once there are more than a handful of numeric data values it is often useful to step back and look at the _distribution_ of the data values: * Where is the bulk of the data located? * Is there a single area of concentration or are there several? * Is the data distribution symmetric or is it skewed, i.e. trails off more slowly in one direction or another? * Are there extreme, or outlying, values? * Are there any suspicious or impossible values? * Are there gaps in the data? * Is there rounding, e.g. to integer values, or _heaping_, i.e. a few particular values occur very frequently? Plots for visualizing distributions include * Strip plots. * Histograms. * Density plots. * Box plots. * Violin plots. * Swarm plots. * Density ridges ## Strip Plots ### Strip Plot Basics A variant of the dot plot is known as a _strip plot_. A strip plot for the city temperature data is ```{r, fig.height = 2, warning = FALSE, class.source = "fold-hide"} thm <- theme_minimal() + theme(text = element_text(size = 16)) ggplot(citytemps) + geom_point(aes(x = temp, y = "All")) + thm + theme(axis.title.y = element_blank(), axis.text.y = element_blank()) ``` The strip plot can reveal gaps and outliers. After looking at the plot we might want to examine the high and low values: ```{r} filter(citytemps, temp > 85) filter(citytemps, temp < 10) ``` For the eruption durations in the `faithful` data a strip plot shows the two modes around 2 and 4 minutes: ```{r, fig.height = 2, warning = FALSE, class.source = "fold-hide"} ggplot(faithful) + geom_point(aes(x = eruptions, y = "All")) + thm + theme(axis.title.y = element_blank(), axis.text.y = element_blank()) ``` ### Multiple Groups Strip plots are most useful for showing subsets corresponding to a categorical variable. A strip plot for the yields for different varieties in the barley data is ```{r, class.source = "fold-hide"} ggplot(barley) + geom_point(aes(x = yield, y = variety)) + theme_minimal() + thm ``` ### Scalability Scalability in this form is limited due to over-plotting. A simple strip plot of `price` within the different `cut` levels in the `diamonds` data is not very helpful: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_point(aes(x = price, y = cut)) + thm + theme(axis.title.y = element_blank()) ``` Several approaches are available to reduce the impact of over-plotting: * reduce the point size; * random displacement of points, called _jittering_; * making the points translucent, or _alpha blending_. Combining all three for examining `price` within `cut` for the `diamonds` data produces ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_point(aes(x = price, y = cut), size = 0.2, position = position_jitter(width = 0), alpha = 0.2) + thm + theme(axis.title.y = element_blank()) ``` Skewness of the price distributions can be seen in this plot, though other approaches will show this more clearly. A peculiar feature reveled by this plot is the gap below 2000. Examining the subset with `price < 2000` shows the gap is roughly symmetric around 1500: ```{r, class.source = "fold-hide"} ggplot(filter(diamonds, price < 2000)) + geom_point(aes(x = price, y = cut), size = 0.2, position = position_jitter(width = 0), alpha = 0.2) + thm + theme(axis.title.y = element_blank()) ``` A plot along these lines was used on the New York Times [front page for February 21, 2021](`r IMG("NYT-2021-02-21.jpeg")`). ```{r, echo = FALSE, out.width = "55%"} knitr::include_graphics(IMG("NYT-2021-02-21.jpeg")) ``` ### Some Notes * With a good combination of point size choice, jittering, and alpha blending the strip plot for groups of data can scale to several hundred thousand observations and ten to twenty of groups. * For very large datat sets it can be useful to look at a strip plot of a sample of the data. * Strip plots can reveal gaps, outliers, and data outside of the expected range. * Skewness and multi-modality can be seen, but other visualizations show these more clearly. * Storage needed for vector graphics images grows linearly with the number of observations. * Base graphics provides `stripchart` and lattice provides `stripplot`. ## Histograms ### Histogram Basics Historams are constructed by binning the data and counting the number of observations in each bin. The objective is usually to visualize the shape of the distribution. The number of bins needs to be * small enough to reveal interesting features; * large enough not to be too noisy. A very small bin width can be used to look for rounding or heaping. Common choices for the vertical scale are: * bin counts, or frequencies; * counts per unit, or densities. The count scale is more intepretable for lay viewers. The density scale is more suited for comparison to mathematical density models. Constructing histograms with unequal bin widths is possible but rarely a good idea. ### Histograms in R There are many ways to plot histograms in R: * the `hist()` function in the base `graphics` package; * `truehist()` in package `MASS`; * `histogram()` in package `lattice`; * `geom_histogram()` in package `ggplot2`. A histogram of eruption durations for another data set on Old Faithful eruptions, this one from package `MASS`: ```{r, message = TRUE, class.source = "fold-hide"} data(geyser, package = "MASS") ggplot(geyser) + geom_histogram(aes(x = duration)) + thm ``` The default settings using `geom_histogram` are less than ideal. Using a binwidth of 0.5 and customized `fill` and `color` settings produces a better result: ```{r, class.source = "fold-hide"} ggplot(geyser) + geom_histogram(aes(x = duration), binwidth = 0.5, fill = "grey", color = "black") + thm ``` Reducing the bin width shows an interesting feature: ```{r, class.source = "fold-hide"} ggplot(geyser) + geom_histogram(aes(x = duration), binwidth = 0.05, fill = "grey", color = "black") + thm ``` * Eruptions were sometimes classified as _short_ or _long_; these were coded as 2 and 4 minutes. * For many purposes this kind of heaping or rounding does not matter. * It would matter if we wanted to estimate means and standard deviations of the durations of the long and short eruptions. * More data and information about geysers is available at https://geysertimes.org/. * For exploration there is no one "correct" bin width or number of bins. * It would be very useful to be able to change this parameter interactively. ### Superimposing a Density A histogram can be used to compare the data distribution to a theoretical model, such as a normal distribution. This requires using a _density scale_ for the vertical axis. The `Galton` data frame in the `HistData` package is one of several data sets used by Galton to study the heights of parents and their children. Adding a normal density curve to a `ggplot` histogram involves: * computing the parameters of the density; * creating the histogram with a density scale using the computed variable `after_stat(density)`; * adding the function curve using `geom_function`, `stat_function`, or `geom_line`. Create the histogram with a density scale using the _computed varlable_ `after_stat(density)`: ```{r galton-hist, eval = FALSE} data(Galton, package = "HistData") ggplot(Galton) + geom_histogram(aes(x = parent, y = after_stat(density)), binwidth = 1, fill = "grey", color = "black") + thm ``` ```{r galton-hist, echo = FALSE} ``` Then compute the mean and standard deviation and add the normal density curve: ```{r galton-hist-dens, eval = FALSE} data(Galton, package = "HistData") p_mean <- mean(Galton$parent) p_sd <- sd(Galton$parent) p_dens <- function(x) dnorm(x, p_mean, p_sd) ggplot(Galton) + geom_histogram(aes(x = parent, y = after_stat(density)), binwidth = 1, fill = "grey", color = "black") + geom_function(fun = p_dens, color = "red") + thm ``` ```{r galton-hist-dens, echo = FALSE} ``` ### Multiple Groups Faceting works well for showing comparative histograms for multiple groups. Histograms of `price` within `cut` for the `diamonds` data: ```{r, fig.width = 8, class.source = "fold-hide"} ggplot(diamonds) + geom_histogram(aes(x = price), binwidth = 1000, color = "black", fill = "grey") + facet_wrap(~ cut) + thm ``` These histograms show counts on the vertical axis, and their sizes reflect the total counts for the groups. Together the plots represent a view of the joint distribution of `cut` and `price`. Switching to a density scale by using `after_stat(density)` for the `y` aesthetic allows the conditional distributions of `price` within groups to be compared: ```{r, fig.width = 8, class.source = "fold-hide"} p <- ggplot(diamonds) + geom_histogram(aes(x = price, y = after_stat(density)), binwidth = 1000, color = "black", fill = "grey") + thm p + facet_wrap(~ cut) ``` By mapping the `fill` aesthetic to `cut` it is possible to produce a stacked histogram or a superimposed histogram * `position = "stack"`, the default, for stacked; * `position = "identity"` for superimposed. But neither works very well visually. For comparing locations of features it can help to facet with a single column. But this may create aspect ratios that are not ideal. ```{r hist-facet-one-col, echo = FALSE, fig.height = 7} p + facet_wrap(~ cut, ncol = 1) + coord_fixed(1.5 * 1e7) ``` ### Scalability Histograms scale very well. * The visual performance does not deteriorate with increasing numbers of observations. * The computational effort needed is linear in the number of observations. * The amount of storage needed for an image object is linear in the number of bins. ## Density Plots ### Density Plot Basics Density plots can be thought of as plots of smoothed histograms. The smoothness is controlled by a _bandwidth_ parameter that is analogous to the histogram binwidth. Most density plots use a [_kernel density estimate_](https://en.wikipedia.org/wiki/Kernel_density_estimation), but there are other possible strategies; qualitatively the particular strategy rarely matters. A density plot of the `geyser` `duration` variable with default bandwidth: ```{r, class.source = "fold-hide"} ggplot(geyser) + geom_density(aes(x = duration)) + thm ``` Using a smaller bandwidth shows the heaping at 2 and 4 minutes: ```{r, class.source = "fold-hide"} ggplot(geyser) + geom_density(aes(x = duration), bw = 0.05) + thm ``` For a moderate number of observations a useful addition is a jittered _rug plot_: ```{r, class.source = "fold-hide"} ggplot(geyser) + geom_density(aes(x = duration)) + geom_rug(aes(x = duration, y = 0), position = position_jitter(height = 0)) + thm ``` ### Scalability Visual scalability is very good. For the `diamonds` data `price` variable: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price)) + thm ``` Density estimates are generally computed at a grid of points and interpolated. Defaults in R vary from 50 to 512 points. Computational effort for a density estimate at a point is proportional to the number of observations. Storage needed for an image is proportional to the number of points where the density is estimated. ### Multiple Groups Density estimates for several groups can be shown in a single plot by mapping a group index to an aesthetic, such as `color`: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density(aes(x = yield, color = site)) + thm ``` Using `fill` and `alpha` can also be useful: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density(aes(x = yield, fill = site), alpha = 0.2) ``` Multiple densities in a single plot works best with a smaller number of categories, say 2 or 3: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density(aes(x = yield, fill = year), alpha = 0.4) + thm ``` Using small multiples, or faceting, may be a better option: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density(aes(x = yield)) + facet_wrap(~ site) + thm ``` These ideas can be combined: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density(aes(x = yield, color = year)) + facet_wrap(~ site) + thm ``` These plots again show lower yields for 1932 than for 1931 for all sites except Morris. Density plots default to using the density scale. For the diamonds data a density plot of `price` faceted on `cut` shows the conditional distributions of `price` at the different `cut` levels: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price)) + facet_wrap(~ cut) + thm ``` Mapping the `y` aesthetic to `after_stat(count)` shows the joint distribution of `price` and `cut`: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price, y = after_stat(count))) + facet_wrap(~ cut) + thm ``` A stacked density plot is sometimes useful but often hard to read: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price, y = after_stat(count), fill = cut), position = "stack") + thm ``` An intermediate option: A faceted plot on the count scale with a muted plot for the full data to allow proportions of the whole to be assessed: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price, y = after_stat(count)), fill = "lightgrey", color = NA, data = mutate(diamonds, cut = NULL)) + geom_density(aes(x = price, y = after_stat(count), fill = cut), position = "stack", color = NA) + facet_wrap(~ cut) + scale_fill_viridis_d(guide = "none") + thm ``` A filled density plot provides a vew of the conditional distribution of `cut` at the different price levels: ```{r, class.source = "fold-hide"} ggplot(diamonds) + geom_density(aes(x = price, y = after_stat(count), fill = cut), position = "fill") + ylab(NULL) + thm ``` This is called a _CD plot_, or a _conditional density plot_. ### Some Notes Computations are generally done with the base R function `density`. `plot` has a method for the results returned by this function, so a density plot can be created with ```{r, eval = FALSE} plot(density(geyser$duration)) ``` The `lattice` package provides the function `densityplot`. ### Interactive Bandwidth Choice Being able to chose the bandwidth of a density plot, or the binwidth of a histogram, interactively is useful for exploration. One way to do this in R (which unfortunately does not work on the RStudio server): ```{r, eval = FALSE} data(geyser, package = "MASS") source("https://stat.uiowa.edu/~luke/classes/STAT7400/examples/tkdens.R") tkdens(geyser$duration, tkrplot = TRUE) ``` Another option: ```{r, eval = FALSE} data(geyser, package = "MASS") source("https://stat.uiowa.edu/~luke/classes/STAT7400/examples/shinydens.R") shinyDens(geyser$duration) ``` ## Boxplots _Boxplots_, or _box-and-whisker_ plots, provide a skeletal representation of a distribution. They are very well suited for showing distributions for multiple groups. There are many variations of boxplots: * Most start with a box from the first to the third quartiles and divided by the median. * The simplest form then adds a whisker from the lower quartile to the minimum and from the upper quartile to the maximum. * More common is to draw the upper whisker to the largest point below the upper quartile $+ 1.5 * IQR$, and the lower whisker analogously. * _Outliers_ falling outside the range of the whiskers are then drawn directly: ```{r, class.source = "fold-hide"} library(gapminder) library(ggplot2) ggplot(gapminder) + geom_boxplot(aes(x = continent, y = gdpPercap)) + xlab(NULL) + thm ``` There are variants that distinguish between _mild outliers_ and _extreme outliers_. A common variant is to show an approximate 95% confidence interval for the population median as a _notch_: ```{r, class.source = "fold-hide"} ggplot(gapminder) + geom_boxplot(aes(x = continent, y = gdpPercap), notch = TRUE) + xlab(NULL) + thm ``` Another variant is to use a width proportional to the square root of the sample size to reflect the strength of evidence in the data: ```{r, class.source = "fold-hide"} ggplot(gapminder) + geom_boxplot(aes(x = continent, y = gdpPercap), notch = TRUE, varwidth = TRUE) + xlab(NULL) + thm ``` With moderate sample sizes it can be useful to super-impose the original data, perhaps with jittering and alpha blending. The outliers in the box plot can be turned off with `outlier.color = NA` so they are not shown twice: ```{r, class.source = "fold-hide"} p <- ggplot(gapminder) + geom_boxplot(aes(x = continent, y = gdpPercap), notch = TRUE, varwidth = TRUE, outlier.color = NA) + xlab(NULL) + thm p + geom_point(aes(x = continent, y = gdpPercap), position = position_jitter(width = 0.1), alpha = 0.1) ``` ## Violin Plots A variant of the boxplot is the _violin plot_: > Hintze, J. L., Nelson, R. D. (1998), "Violin Plots: A Box > Plot-Density Trace Synergism," _The American Statistician_ 52, > 181-184. The violin plot uses density estimates to show the distributions: ```{r, class.source = "fold-hide"} ggplot(gapminder) + geom_violin(aes(x = continent, y = gdpPercap)) + xlab(NULL) + thm ``` By default the "violins" are scaled to have the same area. They can also be scaled to have the same maximum height or to have areas proportional to sample sizes. This is done by adding * `scale = "width"` or * `scale = "count"` to the `geom_violin` call. A comparison of boxplots and violin plots: ```{r, class.source = "fold-hide"} ggplot(gapminder) + geom_boxplot(aes(x = continent, y = gdpPercap)) + geom_violin(aes(x = continent, y = gdpPercap), fill = NA, scale = "width", linetype = 2) + xlab(NULL) + thm ``` A combination of boxplots and violin plots: ```{r, class.source = "fold-hide"} ggplot(gapminder) + geom_violin(aes(x = continent, y = gdpPercap), scale = "width") + geom_boxplot(aes(x = continent, y = gdpPercap), width = .1) + xlab(NULL) + thm ``` There are other variations, e.g. _vase plots_. Boxplots do not reflect the shape of a distribution. For the `eruptions` in the `faithful` data set: ```{r, class.source = "fold-hide"} ggplot(faithful) + geom_boxplot(aes(y = eruptions, x = "Box")) + geom_violin(aes(y = eruptions, x = "Violin")) + xlab(NULL) + thm ``` ## Swarm Plots Swarm plots show the full data in a form that also shows the density. There are a number of variations and names, including _beeswarm plots_, _violin scatterplots_, _violin strip charts_, and _sina plots_ [Sina plots](https://www.tandfonline.com/doi/full/10.1080/10618600.2017.1366914) are available as `geom_sina` in the `ggforce` package: ```{r, class.source = "fold-hide"} library(ggforce) ggplot(gapminder, aes(x = continent, y = gdpPercap)) + geom_sina(size = 0.2) + xlab(NULL) + thm ``` Combined with a width-scaled violin plot: ```{r, class.source = "fold-hide"} ggplot(gapminder, aes(x = continent, y = gdpPercap)) + geom_violin(scale = "width") + geom_sina(color = "blue", size = 0.4, scale = FALSE) + xlab(NULL) + thm ``` ## Effectiveness and Scalability * Boxplots are very simple and easy to compare. * Boxplots strongly emphasize the middle half of the data. * Boxplots may not be easy for a lay viewer to understand. * Box plots scale fairly well visually and computationally in the number of observations; over-plotting/storage of outliers becomes an issue for larger data sets * Violin plots scale well both visually and computationally in the number of observations. ```{r, fig.width = 11, fig.height = 4, class.source = "fold-hide"} library(patchwork) p1 <- ggplot(diamonds) + geom_boxplot(aes(x = cut, y = price)) + xlab(NULL) + thm p2 <- ggplot(diamonds) + geom_violin(aes(x = cut, y = price)) + xlab(NULL) + thm p1 + p2 ``` * Scalability in the number of cases for swarm or sina plots is more limited. * The number of groups that can be handled for comparison by these plots is in the range of a few dozen. ```{r, fig.width = 11, fig.height = 5, class.source = "fold-hide"} library(lattice) p1 <- ggplot(barley) + geom_boxplot(aes(x = site, y = yield, fill = year)) + xlab(NULL) + thm p2 <- ggplot(barley) + geom_violin(aes(x = site, y = yield, fill = year)) + xlab(NULL) + thm p1 + p2 ``` Axes can be flipped to avoid overplotting of labels: ```{r, fig.width = 11, class.source = "fold-hide"} library(lattice) p3 <- p1 + coord_flip() + guides(fill = "none") p4 <- p2 + coord_flip() p3 + p4 ``` Faceting can also be used to arrange groups of boxplots or violin plots. For life expectancy by continent over the years in the `gapminder` data: ```{r, class.source = "fold-hide"} library(dplyr) ggplot(filter(gapminder, year %% 10 == 2, continent != "Oceania")) + geom_boxplot(aes(x = lifeExp, y = factor(year))) + facet_wrap(~ continent, ncol = 1) + theme_minimal() + theme(text = element_text(size = 12)) + theme(strip.text.x = element_text(hjust = 0)) + ylab(NULL) ``` A related visualization motivated by a graph in the Economist is available [here](https://cinc.rud.is/web/packages/ggeconodist/). ## Ridgeline Plots [Ridgeline plots](https://blog.revolutionanalytics.com/2017/07/joyplots.html), also called _ridge plots_ or _joy plots_, are another way to show density estimates for a number of groups that has become popular recently. The package `ggridges` defines `geom_density_ridges` for creating these plots: ```{r, message = FALSE, class.source = "fold-hide"} library(ggridges) ggplot(barley) + geom_density_ridges(aes(x = yield, y = site, group = site)) + ylab(NULL) + thm ``` Grouping by an interaction with a categorical variable, `year`, produces separate density estimates for each level. Mapping the `fill` aesthetic to `year` allows the separate densities to be identified: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density_ridges( aes(x = yield, y = site, group = interaction(year, site), fill = year)) + ylab(NULL) + thm ``` Alpha blending may sometimes help: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density_ridges( aes(x = yield, y = site, group = interaction(year, site), fill = year), alpha = 0.8) + ylab(NULL) + thm ``` Adjusting the vertical scale may also help: ```{r, class.source = "fold-hide"} ggplot(barley) + geom_density_ridges( aes(x = yield, y = site, group = interaction(year, site), fill = year), scale = 0.8) + ylab(NULL) + thm ``` Sometimes reordering the grouping variable, `year` in this case, can help. The factor levels of `year` can be reordered to match the order of average yealds within each year by ```{r, eval = FALSE} reorder(year, yield) ``` Using `-yield` produces the reverse order. ```{r, message = FALSE, class.source = "fold-hide"} library(dplyr) ggplot(mutate(barley, year = reorder(year, -yield))) + geom_density_ridges(aes(x = yield, y = site, group = interaction(year, site), fill = year), scale = 0.8) + ylab(NULL) + thm ``` With some tuning ridgeline plots can scale well to many distributions. An example from [Claus Wilke's book](https://clauswilke.com/dataviz/): The `ggplot2movies` package provides data from [IMDB](https://imdb.com/) on a large number of movies, including their lengths, in a tibble `movies`: ```{r} library(ggplot2movies) dim(movies) head(movies) ``` A ridgeline plot of the movie lengths for each year: ```{r, message = FALSE, class.source = "fold-hide"} library(dplyr) mv12 <- filter(movies, year > 1912) ggplot(mv12, aes(x = length, y = year, group = year)) + geom_density_ridges(scale = 10, size = 0.25, rel_min_height = 0.03) + scale_x_continuous(limits = c(0, 200)) + scale_y_reverse(breaks = c(2000, 1980, 1960, 1940, 1920)) + theme_minimal() ``` This shows that since the early 1960's feature film lengths have stabilized to a distribution centered around 90 minutes: Another nice example: [DW-NOMINATE](https://en.wikipedia.org/wiki/NOMINATE_%28scaling_method%29) scores for measuring political position of members of congress over the years: ```{r, echo = FALSE, out.width = "70%"} knitr::include_graphics(IMG("polarization.jpeg")) ``` [Original code]( http://rpubs.com/ianrmcdonald/293304) by Ian McDonald; another version is provided in [Claus Wilke's book](https://clauswilke.com/dataviz/). ## Reading Chapter [_Visualizing distributions: Histograms and density plots_](https://clauswilke.com/dataviz/histograms-density-plots.html) in [_Fundamentals of Data Visualization_](https://clauswilke.com/dataviz/). Section [_Histograms and density plots_](https://socviz.co/groupfacettx.html#histograms) in [_Data Visualization_](https://socviz.co/). Chapter [_Visualizing data distributions_](https://rafalab.dfci.harvard.edu/dsbook/distributions.html) in [_Introduction to Data Science Data Analysis and Prediction Algorithms with R_](https://rafalab.dfci.harvard.edu/dsbook/). ## Interactive Tutorial An interactive [`learnr`](https://rstudio.github.io/learnr/) tutorial for these notes is [available](`r WLNK("tutorials/dists.Rmd")`). You can run the tutorial with ```{r, eval = FALSE} STAT4580::runTutorial("dists") ``` You can install the current version of the `STAT4580` package with ```{r, eval = FALSE} remotes::install_gitlab("luke-tierney/STAT4580") ``` You may need to install the `remotes` package from CRAN first. ## Exercises 1. Consider the code ```{r, eval = FALSE} library(ggplot2) data(Galton, package = "HistData") ggplot(Galton, aes(x = parent)) + geom_histogram(---, fill = "grey", color = "black") ``` Which of the following replacements for `---` produces a histogram with bins that are one inch wide and start at whole integers? a. `binwidth = 1` b. `binwidth = 1, center = 66.5` c. `binwidth = 2, center = 66` d. `center = 66` 2. Consider the code ```{r, eval = FALSE} library(ggplot2) ggplot(faithful, aes(x = eruptions)) + geom_density(---) ``` Which of the following replacements for `---` produces a density plot with the area under the density in blue and no black border? a. `color = "lightblue"` b. `fill = "black", color = "lightblue"` c. `fill = "lightblue", color = NA` d. `fill = NA, color = "black"` 3. Consider the code ```{r, eval = FALSE} library(ggplot2) library(gapminder) p <- ggplot(gapminder, aes(y = continent, x = lifeExp)) ``` Which of the following produces violin plots without trimming at the smallest and largest observations, and including a line at the median? a. `p + geom_violin(trim = FALSE)` b. `p + geom_violin(trim = TRUE, show_median = TRUE)` c. `p + geom_violin(trim = FALSE, draw_quantiles = 0.5)` d. `p + geom_violin(trim = TRUE, show_quantiles = 0.5)` 4. Density ridges can also show quantiles, but the details of how to request this are different. Consider this code: ```{r, eval = FALSE} library(ggplot2) library(ggridges) library(gapminder) ggplot(gapminder, aes(x = lifeExp, y = year, group = year)) + geom_density_ridges(---) ``` Which of the following replacements for `---` produces density ridges with lines showing the locations of the medians? a. `quantiles = 0.5` b. `quantile_lines = TRUE, quantiles = 0.5` c. `quantile_lines = TRUE` d. `draw_quantiles = 0.5`